Connecting eye rig to head bone
You might need to adjust how you are trying to parent objects in order to solve the issue you are having.

When you want to parent an object to a bone that you plan to move with pose mode, you should try to parent it to the bone, rather than parent to the armature object. 



Here is a simple workflow that should work for your situation.


Select the armature that you want to follow the bone in another armature
Now select the armature that contains the bone that you wish to parent to.
Use the parenting menu and select bone, or bone relative, depending on whether you want your child armature to retain its offset or not.


Hope that helps, happy blendering!

Note that:

If the armature containing the bone you want to parent to has multiple bones, you will need to select the proper one before doing so.

You can do that by going into pose mode prior to parenting, choose the bone that is to be the parent, then go back to object mode.